U nit 2: P robability and distributions L ecture 2: N ormal distribution S tatistics 101 Mine C ¸ etinkaya-Rundel September 17, 2013
Normal distribution Normal distribution 1 Normal distribution model 68-95-99.7 Rule Standardizing with Z scores Percentiles Recap 2 Evaluating the normal approximation Application exercises 3 Finding probabilities // Quality control Finding cutoff points // Hot bodies Conditional probability // SAT scores Finding missing parameters // Auto insurance premiums Statistics 101 U2 - L2: Normal distribution Mine C ¸ etinkaya-Rundel
Normal distribution Heights of males http://blog.okcupid.com/index.php/the-biggest-lies-in-online-dating/ Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 2 / 26
Normal distribution Heights of males “The male heights on OkCupid very nearly follow the expected normal distribution – except the whole thing is shifted to the right of where it should be. Almost universally guys like to add a couple inches.” “You can also see a more subtle vanity at work: starting at roughly 5’ 8”, the top of the dotted curve tilts even further rightward. This means that guys as they get closer to six feet round up a bit more than usual, stretching for that coveted psychological benchmark.” http://blog.okcupid.com/index.php/the-biggest-lies-in-online-dating/ Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 2 / 26
Normal distribution Heights of females http://blog.okcupid.com/index.php/the-biggest-lies-in-online-dating/ Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 3 / 26
Normal distribution Heights of females “When we looked into the data for women, we were surprised to see height exaggeration was just as widespread, though without the lurch towards a benchmark height.” http://blog.okcupid.com/index.php/the-biggest-lies-in-online-dating/ Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 3 / 26
Normal distribution Normal distribution model Normal distribution Denoted as N ( µ, σ ) → Normal with mean µ and standard deviation σ Unimodal and symmetric, bell shaped curve, that also follows very strict guidelines about how variably the data are distributed around the mean Therefore while most variables are nearly normal, but none are exactly normal Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 4 / 26
Normal distribution 68-95-99.7 Rule 68-95-99.7 Rule 68% 95% 99.7% µ − 3 σ µ − 2 σ µ + 2 σ µ + 3 σ µ − σ µ µ + σ For nearly normally distributed data, Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 5 / 26
Normal distribution 68-95-99.7 Rule 68-95-99.7 Rule 68% 95% 99.7% µ − 3 σ µ − 2 σ µ + 2 σ µ + 3 σ µ − σ µ µ + σ For nearly normally distributed data, about 68% falls within 1 SD of the mean, Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 5 / 26
Normal distribution 68-95-99.7 Rule 68-95-99.7 Rule 68% 95% 99.7% µ − 3 σ µ − 2 σ µ + 2 σ µ + 3 σ µ − σ µ µ + σ For nearly normally distributed data, about 68% falls within 1 SD of the mean, about 95% falls within 2 SD of the mean, Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 5 / 26
Normal distribution 68-95-99.7 Rule 68-95-99.7 Rule 68% 95% 99.7% µ − 3 σ µ − 2 σ µ + 2 σ µ + 3 σ µ − σ µ µ + σ For nearly normally distributed data, about 68% falls within 1 SD of the mean, about 95% falls within 2 SD of the mean, about 99.7% falls within 3 SD of the mean. Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 5 / 26
Normal distribution 68-95-99.7 Rule 68-95-99.7 Rule 68% 95% 99.7% µ − 3 σ µ − 2 σ µ + 2 σ µ + 3 σ µ − σ µ µ + σ For nearly normally distributed data, about 68% falls within 1 SD of the mean, about 95% falls within 2 SD of the mean, about 99.7% falls within 3 SD of the mean. It is possible for observations to fall 4, 5, or more standard deviations away from the mean, but these occurrences are very rare if the data are nearly normal. Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 5 / 26
Normal distribution 68-95-99.7 Rule Describing variability using the 68-95-99.7 Rule SAT scores are distributed nearly normally with mean 1500 and standard deviation 300. Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 6 / 26
Normal distribution 68-95-99.7 Rule Describing variability using the 68-95-99.7 Rule SAT scores are distributed nearly normally with mean 1500 and standard deviation 300. 68% 95% 99.7% 600 900 1200 1500 1800 2100 2400 Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 6 / 26
Normal distribution 68-95-99.7 Rule Describing variability using the 68-95-99.7 Rule SAT scores are distributed nearly normally with mean 1500 and standard deviation 300. 68% 95% 99.7% 600 900 1200 1500 1800 2100 2400 ∼ 68% of students score between 1200 and 1800 on the SAT. Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 6 / 26
Normal distribution 68-95-99.7 Rule Describing variability using the 68-95-99.7 Rule SAT scores are distributed nearly normally with mean 1500 and standard deviation 300. 68% 95% 99.7% 600 900 1200 1500 1800 2100 2400 ∼ 68% of students score between 1200 and 1800 on the SAT. ∼ 95% of students score between 900 and 2100 on the SAT. Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 6 / 26
Normal distribution 68-95-99.7 Rule Describing variability using the 68-95-99.7 Rule SAT scores are distributed nearly normally with mean 1500 and standard deviation 300. 68% 95% 99.7% 600 900 1200 1500 1800 2100 2400 ∼ 68% of students score between 1200 and 1800 on the SAT. ∼ 95% of students score between 900 and 2100 on the SAT. ∼ 99.7% of students score between 600 and 2400 on the SAT. Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 6 / 26
Normal distribution 68-95-99.7 Rule Participation question A doctor collects a large set of heart rate measurements that approx- imately follow a normal distribution. He only reports 3 statistics, the mean = 110 beats per minute, the minimum = 65 beats per minute, and the maximum = 155 beats per minute. Which of the following is most likely to be the standard deviation of the distribution? (a) 5 (b) 15 (c) 35 (d) 90 Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 7 / 26
Normal distribution 68-95-99.7 Rule Participation question A doctor collects a large set of heart rate measurements that approx- imately follow a normal distribution. He only reports 3 statistics, the mean = 110 beats per minute, the minimum = 65 beats per minute, and the maximum = 155 beats per minute. Which of the following is most likely to be the standard deviation of the distribution? (a) 5 → 110 ± (3 × 5) = (95 , 125) (b) 15 → 110 ± (3 × 15) = (65 , 155) (c) 35 → 110 ± (3 × 35) = (5 , 215) (d) 90 → 110 ± (3 × 90) = ( − 160 , 380) Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 7 / 26
Normal distribution Standardizing with Z scores SAT scores are distributed nearly normally with mean 1500 and stan- dard deviation 300. ACT scores are distributed nearly normally with mean 21 and standard deviation 5. A college admissions officer wants to determine which of the two applicants scored better on their stan- dardized test with respect to the other test takers: Pam, who earned an 1800 on her SAT, or Jim, who scored a 24 on his ACT? Jim Pam 600 900 1200 1500 1800 2100 2400 6 11 16 21 26 31 36 Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 8 / 26
Normal distribution Standardizing with Z scores Standardizing with Z scores Since we cannot just compare these two raw scores, we instead compare how many standard deviations beyond the mean each observation is. Pam’s score is 1800 − 1500 = 1 standard deviation above the mean. 300 Jim’s score is 24 − 21 = 0 . 6 standard deviations above the mean. 5 Jim Pam −2 −1 0 1 2 Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 9 / 26
Normal distribution Standardizing with Z scores Standardizing with Z scores (cont.) These are called standardized scores, or Z scores . Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 10 / 26
Normal distribution Standardizing with Z scores Standardizing with Z scores (cont.) These are called standardized scores, or Z scores . Z score of an observation is the number of standard deviations it falls above or below the mean. Z scores Z = observation − mean SD Statistics 101 (Mine C ¸ etinkaya-Rundel) U2 - L2: Normal distribution September 17, 2013 10 / 26
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